February 6,27 LECTURE #8 Nn-Symmrphic Space Grups
42 CHAPTER 2 SYMMETRY DESCRPTON OF CRYSTALS + + + + + Fig 2- a r: i,_ + -'-- 2 -'--'- / / / / -@ X~t~~,/?\~\\~ -'--' - / + ~ Fig2-lb C2h\P2t/b) ~f~ C\~ ~Y-eV lsl't The 73 symmrphic space grups that ~ generated ill tris R'laRRir ~ ~btul fr all crystal systems The nternatinal ntatin displays rather clearly the symmetry infrmatin, while the Schenflies symbl is just the pint grup symbl with a fairly arbitrary numerical superscript t index each f the space grups with a given pint grup Ntice that the nternatinal symbls fr the symmrphic space grups cntain n glide plane r screw axis symbls since these are nt essential t describe the space grup; nevertheless, they may be f!enerated when the pint peratins are multiplied (cmbined) by the unit cell translatins r the centering cnditins 2-6c Nnsymmrphic space grups The develpment f the nnsymmrphic space grups fllws in the same general manner as symmrphic space grups, but the additin f glide planes and screw axes makes the prblem much mre cmplex We shall nt shw hw they may be develped (see the ntes) but we give tw examples f the results Figure 2-11 a shws ne f the rthrhmbic nnsymmrphic space grups t has similarities t P222 but it als has differences \ (!\1":~V've l:~,\e '~~ t(~\c\ 11'\ cc\ \'(!)1S (~avt\ (') C ~1 ) ~ k ~ ~ '1- i '- v\ \: (cq ~t ~~ &t'~ ~! Q ( ~ ~ OY\~ &t~~\q(e\m~"t t:t-lc)114 t~e Ct'(t'~ 1t"Q~ -\: b e \~ e V ~ \f"w\ ed
The half arrws pinting in the a- and b-directins indicate 2J screw axes These peratins imply a C2(2) rtatin abut each f the screw axes indicated and then a translatin by a/2 r b/2, as indicated by the directin f the half arrw By fllwing the effects f these peratins n the circles we can see that all the screw axes and twfld axes are, indeed, symmetry peratins The 21 screw axes d nt intersect the twfld axes; thus, thc arrangement f circles, which indicates general equivalent psitins, is rather different frm that in the P222 space grup Nevertheless, bth f these space grups have fur general equivalent psitins n Fig 2-11 b a mnclinic, nnsymmrphic space grup is shwn that has 21 screw axes in the c-directin as well as a glide plane parallel t the page at c/4 The cmbinatin f the b-glide and 21-axes prduces centers f inversin, which are indicated by the very small circles The ntatin that is used fr space grup diagrams is summarized in Table 2-4 26d nternatinal ntatin fr planes and axes Table 2-4 shws the nternatinal space grup symbls fr symmetry planes, and Table 2-5 shws the crrespnding symbls fr symmetry axes Figure 2-11 b has a hrizntal glide plane at {u(o,, c/4) '1( b/2, O)} and screw axes f the type {C2(,, 1) '1(,, c/2) J Screw axes in the ab-plane can be fund in Figure 2-11 a As can be seen frm Table 2-5, using the nternatinal ntatin, an na screw axis can be written as {n[o,, 1] '1(,, ac/n)} Fr example, successive peratins f a 65 screw give pints at c =, 5c/6, 1c/6, lsc/6, 2c/6, 2Sc/6 T bring these back int ne unit cell 6c/6 = c can always be subtracted Thus, we have c =, 5c/6, 4c/6 = 2c/3, 3c/6 = c/2, 2c/6 = c/3, c/6 Figure 2-12 shws all the pssible screw axes peratins Ntice hw the 65 peratin is enantimrphic t the 61, peratin
Symbl m a,b Table 2-4 Symbls f the symmetry planes (frm the nternatinal Tables) Graphical symbl Symmetry plane Nrmal t plane Parallel t plane Nature f glide translatin f prjectin,j f prjectin Reflectin Nne (Nan f the plane is at pjane(mirrr) :-1 this is shwn by printins 1 besidethe symbl) ------- -- al2 alng [1]r bl2 alng [1]; r alng (OO) Axial glide c plane - Nne c2a1ngz-axis;r(a+b+c)/2 alng (ll] n rhmbhedral axes n Diagnal glide ------- (a+b)/2r (b+c)/2 r (c+a)/2; plane (net) r r (a+b+c)/2 (tetragnal and ~glide cubic) d Diamnd" ----, (axb)/4 r (b:!:c)/4r(cxa)/4; plane ---- r (a:j;bxc)/4 (tetragnal and i cubic) See nte belw NOTE t n the diamolld" 8ide plane the slide translatin is half nf the ~ultant f the t~ pssible axial 8ide translatins The arrws in the first diagram shw the directin f the hrizntal ~mpnent f the translatin when the z-cmpnent is psitive n tl><secnd diagram the arrw shws the actual difeclln f the slide translaton; there Salways anher diamnd-8ide rejcclin plane parajlelt the firs with a heillu difference f t and with the arrw pintina alns the c diasnal f the cell face Symbl Table 2-5 Symbls f the symmetry axes (frm the nternatinal Tables) Rtatin Nne Nne 4 Rtatin mnad tetrad Nne r nversin Nne 4t Screw - cl4 mnad tetrads 4s 2c/4 2 Rtatin Nne diad (nrmal t paper) 4s 3c/4 " Nature f Graphical Nature f right-handej symbl right-handed ' Symmetry Graphical screw trans- :Symbl Symmetry (nrmal t screw transaxis symbl latin alng axis plane f latin alng the axis paper) the axis (paralleltpaper) --- ~nversin ~Nne tetrad 2, Screw t cl2 diad (nrmal t paper) 6 Rtatin Nne " hexad --- Either (J!UBe' paper) al2 r b/2 6, Screw cl6 hexads Nrmalt 6, 2c/6 paper, 3 Rtatin Nne 6a 3cl6 triad A 6 4c/6 3, Screw -- Acl3 3, triads 6, 2cl3 S~/6 ~nversin A Nne ~nversin ~Nne triad hexad,
+ + 2 4+~ 21 + Fig 2-12 All the pssible crystallgraphic screw peratins 3 1+ 3~ ~+ + $ + + + + 4 + + '\ + -- + + \ t;; +\ 6
-f (b) c J_/ "' '" a - - Ti Fig 1-7 Tw diagrams shwing a unit cell f Ti2' The same idea f translatinal symmetry is expressed by saying that when the atmic arrangement in the crystal is viewed frm any pint r, it is identical when viewed frm the pint, r = r + tm (1-1 Ob) 1-4b Screw peratins Figure 1-7 shws ne unit cell f crystal Ti2 One must imagine that the crystal is made up f these cells repeated again and again ~11Lhe 5et'l!(, f :c~ 1 19a Alld 1 19b t fill all space Nw examine the symmetry peratins Taking the rigin at the center f the unit cell, there are clearly the fllwing eight symmetry peratins: E, C2, 2C2", i, O'h' 2'd' Hwever, as is nw shwn, there are eight mre Cnsider a C4 peratin abut the c-axis fnwed by a translatin c/2 alng the c-axis and a/2 alng the a- and b-axes This is written, in the Seitz ntatin, as where T = a/2 + b/2 + c/2 (1-11) Nte that befre and after this peratin the crystal, nt ne unit cell, but the entire crystal, is in an equivalent psitin Thus, this is a symmetry peratin f the crystal and is called a screw peratin The imprtant pint t nte is that the translatin T is a fractin f the unit cell size (n general, fr screw peratins the fractins are always 1/2, 1/3, 1/4, 1/6, r integral multiplies f the latter three) Altgether there are eight symmetry peratins f the crystal invlving T These are {C4\ T}, {C43 T}, 2{C2' T}, 2{S41 T}, 2{O'v\ T} Thus, this crystal has the 16 symmetry peratins discussed here and, f curse, the infinite number f translatins f the unit cell We shuld a bit niffptpqtly Namely, it is a rtatin fllwed by a translatin a "ractin f a unit cell parallel t the rtatin axis "1A, O"J,, CQt\ be 1: c ~avc C't CC\'\~lat'feLl ~\~""'Mct'{'1 ()~'\\~ \~ \ H,c SLr~\V () ~ Q{CL-\ ~ l') 1/\ S -\:~~~\/\ \ A-t QC cu \A-t
142 ntrductin t Crystallgraphy Ti " Ti FG7-1 The tetragnal Ti2 structure prjected nt (1) Titanium ins are at,,; t,t,t Oxygen ins are at 3,3,; 8,2,t; 7,7,; 2,8, t 7-11 Rutile structure The structure pssessed by rutile, TiOh by cassiterite, Sn2' and by a number f ther substances with small catins is shwn in Fig 7-1 The structure is tetragnal; fr Ti2, a = 4594 A, C= 2958 A; fr Sn2' a = 4737 A:'c-: 3186 A The space grup is P42/mnm, the Ti4+ ins ccupy psitins (2a):,,; t,t,t, and the 2- ins ccupy psitins (4f): ::(x,x,o; t + X,1 - x,t) with x very nearly 3 The titanium in is surrunded by six xygen ins which frm a slightly distrted ctahedrn EXERCSE7-16 Calculate the distance frm the Ti4+ in at 1 1 t in the rutile structure t each f its six 2- neighbrs EXERCSE7-17 Describe the nearest neighbr envirnment f an 2- in in Ti2 Give the distances wherever necessary
-+ c -+t ~ -+ ~ c -+t -+t -+t c ō -+t ~ -< 6 \ 6 V) - M en :: ~ ~- :: --- ~ 'iij c E ~u ;a c: - >- c :: ::s 'iij e :: U en 5 E~ - u ~u u c: -:: - ::s ~u :: c: t'! r- ~ c ::s ọ 1) u u~c en ~C) a:- v x _ ",< : :t (J C c '"C P- ~ Ṇ u:
Cvt,-tV'~~~ \V~'\~ Tv-e ~,"<M\A1\t)\f ~lt\c 2\1\ S ~-\-vuc \u\re SOME SMPLE STRUCTURES 143 "-12 Zinc sulfide s'ruc'ure Zinc blende, ZnS, is cubic The ZnH ins are at,, + face centering, and the S2- ins are at t, t, t + face centering (see Fig 7-11) The space grup is F43m, and the lattice dimensin fr ZnS is 549 A f the zinc and sulfur atms were identical, this wuld be the diamnd structure Each atm in ZnS is surrunded by a regular tetrahedrn f atms f the ppsite type EXERCSE7-18 Calculate the structure factrs, in terms f f, fr the ( t), (2), and (22) planes f cubic ZnS Cmpare with the crrespnding results fr diamnd frm Exercise 7-12 t Zn Zn Zn b 8* 8~ t Zn @ Zn t j z" t Zn Zn 8* 8* FG 7-11 The cubic ZnS structure